1
SOLIDIFICATION BEHAVIOUR OF NATURAL SILICATE MELTS
AND VOLCANOLOGICAL IMPLICATIONS
Gianluca Iezzi
1,2
, Silvio Mollo
2
, Guido Ventura
2
1 Dipartimento di Geotecnologie per l’ambiente ed il territorio, Università G. d’Annunzio, Chieti, Italy.
2 Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy.
Corresponding author: Guido Ventura, Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata 605, 00143
Roma, Italy. Department Seismology and Tectonophysics; Phone (+39) 06-51860221; Fax (+39)06-5041303; e-mail: ventura@ingv.it
ABSTRACT
The thermodynamic and physical properties of magmas have been extensively investigated as a function of T , P , f O
2
and composition allowing the development of accurate phase stability, viscosity, and diffusion models. However, how the silicate melt properties are influenced by kinetic effects is still an open question. The most important transformation of a magma is its solidification due to cooling, i.e. the transition from a silicate melt to a rock. Solidified magmas may be crystalline, vitreous, or a mixture of glass and crystals. If the cooling rate is larger enough to prevent crystallization, a magma can encompass the supercooling region without crystallisation. The smallest cooling rate that suppresses or strongly limited the nucleation of crystals is the critical cooling rate Rc . Melts with low
Rc persist in a metastable liquid state and have a good glass forming ability (GFA). GFA and Rc of melts can be quantitatively estimated using (1) the reduced glass transition parameter Trg = Tg/Tm ( Tg, temperature of glass transition; Tm, temperature of melting), and (2) the viscosity fragility concept. As stated by the theory, strong liquids with high Trg values have good GFA and low Rc , whereas fragile liquids with low Trg have a poor GFA and high R c. Using available experimental data and theoretical models, we analyze the kinetic effects in dry magmas of different composition. The obtained results are relevant for the formation of lava flows and domes. In sub-alkaline magmas, Trg linearly increases and
Rc decreases as the Si and Al content increases. Rc of basalts range between 10
1
and 10
3
K/s. In dacitic and rhyolitic melts, Rc is between 10
-3
and 10
-5
K/s. Alkaline melts have Trg values lower than those of sub-alkaline compositions. Results are consistent with the available experimental data. The sluggish kinetics of nucleation determined by using the relation Rc vs Trg is also in agreement with the experimental and theoretical data for synthetic silicate melts. The outlined solidification behaviour of magmatic melts has a profound influence on the viscosity paths of magmas. Depending on the Trg and
Rc values, less evolved magmas may have a viscosity larger than that of more evolved magmas due to the rapid crystallization induced by the cooling during their flowing on the Earth. The glassy portion of poorly evolved magmas is indicative of rapid cooling, whereas the glassy fraction of evolved magmas is not unequivocally indicative of rapid cooling being their typical Rc values low. Basaltic lavas may flow on the Earth surface for long times only if they have a temperature close to T m
, whereas more evolved lavas can flow for longer periods with temperatures well below T m
. Fully glassy lavas like obsidians have invariably rhyolitic or trachytic compositions.

2
INTRODUCTION
Natural silicate melts solidify on the Earth forming magmatic rocks. Lava flows, which can be considered nearly to fully degassed magmas, represent the most abundant volcanic product on the Earth surface in both subaerial and submarine environments. Lavas contain variable amounts of crystals, virtually from 0 to 100 vol.%. The melt to rock transition generally involves nucleation and crystal growth. However, if the cooling process is rapid enough, nucleation is suppressed and the solidified magmas will be fully glassy (e.g., obsidians). Many studies on natural lavas testify an extreme variability of textures and suggest a different ability of the melt to crystallise (or not) as function of the composition (Donaldson, 1979; Lofgren, 1980). However, a comprehensive model relating composition and facility (or not) to vitrify is not available for natural silicate melts. As an example, the andesitic lava dome of the Soufriere Hills volcano, Montserrat, has about 90 vol.% of crystals, whereas the cogenetic rhyolitic lava dome products are quasi -glassy; this feature has been explained by differences in the nucleation behaviour related to the melt composition (Westrich et al., 1988; Sparks et al., 2000).
The trachytic and rhyolitic obsidians, which occur in many volcanoes, e.g. at Lipari Island (Italy), Coso
Volcanic Field (California), Volcan Las Navajas (Mexico), Obsidian Dome and Little Glass Mountain
(California) and Mayor Island (New Zeland) among many others, have a crystal content lower than 5 vol. % or, more commonly, lower than 1 vol. % (Fink, 1983; Swanson et al., 1989; Nelson et al., 1990;
Crisci et al., 1991; Manley and Bacon, 2000; Yeğingil, et al., 2002; Gottsmann and Dingwell, 2002;
Castro et al., 2002).
In general, more viscous lavas tend to develop a higher flow thickness and move for significant shorter distance with respect to less evolved and viscous lavas (Cas and Wright, 1988). When compared to crystal-poor evolved lavas, basic lavas contain a lower amount of glass. Basaltic lavas from Hawaii emplace as relatively thin and long flows have a crystal content that increases from the outer to the inner portions (Burkhard, 2005 and references therein). The presence of a glassy crust in basaltic lava flows, which is related the the geometry and fluid-dynamic of lava emplacement (Lyman et al., 2005; Cashman et al., 2006), is normally thin and the heat transfer is probably lower of that of thick, silica-rich and moreviscous flow (Manley, 1996). For instance, the interior part of the well studied Obsidian Dome (California; Castro et al. 2002) has a glassy texture (crystals < 5 vol. %).
Alternatively, silicic and relatively crystal-rich lavas as the dome from the Kikai caldera (Japan) has a more crystal-rich outer portion. This feature could be explained by the different silica content, which is of about ~73 wt.% in the inner portion and ~68 wt.% in the outer portion. Mingled lavas also show that the compositionally less evolved fraction is more crystalline, despite the cooling rate of the entire flow or dome is roughly the same (Iezzi and Ventura, 2000; Ventura, 2004). Finally, submarine lavas like basaltic pillows, which are efficiently cooled by sea water have only a thin glassy surface and a crystalrich to holocrystalline interior (Schiffman and Lofgren, 1982). All these observations evidence a general tendency: evolved lava flows are reluctant to nucleate and more able to vitriphy with respect to the less evolved ones.
The development (or not) of nucleation and crystal growth processes in magmas depends on a number of parameters that include: dissolved volatile content, composition, temperature, viscosity, chemical diffusion, pressure, and thermal conductivity (Dowty, 1980; Brandeis et al., 1984; Cashman,
1991; Lasaga, 1997; Blundy et al., 2006). As an example, during the ascent to the surface of volatilebearing magmas, degassing processes induce crystallization (Hammer et al., 1999) because gas

3 exsolution increases the liquidus temperature(s). In this study, the melting temperature data (see below) have been analyzed exclusively under anhydrous and atmospheric pressure and redox conditions.
Our aim is to highlight the relationships between kinetics and composition of natural, anhydrous silicate melts. This analysis is of crucial importance because crystallization processes strongly control the rheological properties of silicate melts (Caricchi et al., 2007 and reference therein). The volume of crystals directly correlates to the viscosity and yield strength, which, in turn, determine the rheological behaviour of a magma (Kerr and Lister, 1991; Pinkerton and Stevenson, 1992; Ishibashi and Sato,
2007). Furthermore, viscosity and yield strength play a major role in the transition from effusive to explosive activity (Gonnerman and Manga, 2007) and in the mechanism of emplacement of lava flows and domes (Griffiths, 2000; Sparks et al., 2000). The results obtained in this study are relevant for the study of cooling, crystallization, and emplacement mechanisms of degassed magmas like lava flows and domes.
THEORETICAL BACKGROUND
Studies on crystal nucleation in simple silicate glasses (CaO–SiO
2
, Li
2
O–BaO–SiO
2
, BaO–
TiO
2
-SiO
2
, MgO–Al
2
O
3
–SiO
2
), or also in other liquid and amorphous materials such as metallic or molecular glasses, demonstrated that several physical parameters are related among them, allowing to model and predict the ability of a silicate liquid to transform in a glass. In other words, the kinetics of transformation of any silicate melt is strongly controlled by its chemical composition (James, 1985;
Sakaguchi, 1995; Weinberg, 1996; Cabral et al., 1997; Lu et al., 2000; Avramov et al., 2003; Cabral et al., 2003; Fan et al., 2005; Fan et al., 2007 and reference therein).
In principle, the ability of a silicate liquid to persist in the amorphous state during cooling is defined as glass-forming ability (GFA) (Avramov et al., 2003; Cabral et al., 2003). GFA can be measured in several ways. The most common way to quantify GFA is the critical cooling rate R c
(also labelled as q cr
), which is the minimum cooling rate at which a liquid can be frozen to a solid glass without crystallization or with a percentage of crystals below 1 vol.% (Cabral et al., 2003; Fokin et al.,
2006). GFA and R c parameters are also able to reveal the facility or difficulty of a silicate melt to nucleate. In homogeneous nucleation processes, a silicate melt with a high GFA will nucleate with difficulty, and conversely, another melt with a high R c will nucleate easily. Therefore, a silicate melt which is reluctant to nucleate is characterized by high GFA and low R c
, and will nucleate prevalently at the surface (e.g. on a sample container; Fokin et al., 2003). On the other hand, surface nucleation alone is also an evidence of slow nucleation of the bulk melt.
An indicative parameter able to link nucleation process and GFA is the time delay of nucleation
(or in general crystallization) named time-lag

(also labelled induction or incubation time).

is defined as the time required to initiate the nucleation at a fixed undercooling, i.e. at static conditions (Fenn,
1977; Dowty, 1980, Lasaga, 1997). From the nucleation theory and in a dynamical cooling regime, the time-lag
 is expected to decrease moving below the liquidus temperature down to low undercooling degrees as a result of the increasing thermodynamic driving force. Therefore, at moderate undercoling
 increases as an effect of the decrease of ion mobility. However, experimental observations commonly report that the time-lag exponentially and monotonically decreased as a function of the undercooling, at least for homogeneous nucleation (Dowty, 1980; Lofgren, 1980; Fokin et al., 2005 and 2006). Timelags between different silicate melt compositions can be compared under a similar degree of undercooling, or by comparing the

amount at the maximum nucleation rate (Fokin et al., 2003). The

4 temperature at which the nucleation rate is maximum is defined
 max
. Fokin et al. (2003) use experimental data and classical nucleation theory (CNT) to demonstrate that the time-lag at
 max
and the maximum nucleation rate I max
are correlated to the macroscopic physic-chemical parameters of a
(simple) silicate melt.

at
 max
is virtually the lowest time-lag having the nucleation function a
Gaussian shape as a function of the undercooling degree. The peak of this distribution is I max
. The position of
 max
is determined by the energetic competition between thermodynamic and kinetic barriers (Lasaga, 1997; Fokin et al., 2003).

at
 max
or I max
is the lower induction time
 min
.
Several parameters have been proposed to predict GFA of silicate melts. According to Tamman
(1904), the higher the melt viscosity at T m
, the lower is the ability to form crystals (Fokin et al., 2003).
In the last two decades, the most valuable parameter used to predict GFA by measuring R c
is the reduced glass transition temperature T rg
although some liquids, e.g. water, cannot be modelled adequately (Lu et al., 2000; Mondal and Murty, 2005; Fan et al., 2007 and reference therein). T rg
is the ratio between the glass transition temperature T g
, which is the temperature at which the glass has a viscosity

10
12
Pa s, and the melting temperature T m
. Therefore, if the value of T rg
is known for a silicate melt, then it is possible to infer GFA of a liquid under supercooling condition.
Both the liquidus and the glass transition temperature of a silicate melt depend on its composition. A parameter able to account for different cation abundances and relative proportions in a silicate melt is the ratio between the number of non-bridging oxygens (NBO) and the number of fourfold coordination cation (T). Normally, Ca, Fe
2+
and Mg, but also Na or K not involved with the charge balancing of trivalent cations, act as network modifiers, whereas Si,
IV
Al,
IV
Ti and possibly
IV
Fe
3+ behave as melt polymerizing cations (e.g. Mysen, 1988; Del Gaudio et al., 2007). As a result, the
NBO/T parameter measures the amount of the network modifier cations and their associated reactive oxygens. This parameter is therefore expected to scale with both T g and T m
, depending on the bulk composition of the silicate melt. NBO/T is virtually 0 for pure SiO
2
, and increases in SiO
2
-poor and/or more alkaline silicates (Mysen, 1988). Further details on the calculation of the NBO/T parameter will be given below.
DATA AND ANALYSIS
Data
A total of 28 different silicate melt compositions have been considered in this study. They have been classified according to the totally alkali vs silica diagram (TAS, Le Maitre et al., 2002). The selected melt compositions are reported in Table 1 and Fig. 1. They cover the majority of natural magma compositions, including sub-alkaline, alkaline and peralkaline melts. The chemical composition of all the samples in Table 1 has been reported from the original studies.
Determination of Tg, Tm and Trg
The melting temperature
 m of the selected compositions have been calculated with the Pele software package (Boudreau, 1999), based on the thermodynamic database of MELTS (Ghiorso and
Sack, 1995; Asimov and Ghiorso, 1998) (Table 1). This code implements a large thermodynamic experimental database and is able to calculate with a reliable confidence several thermodynamic parameters for silicate systems on the basis of free energy minimization.

5
All the measured viscosities as a function of temperatures reported in the original study have been considered. Several other viscosity data at other temperatures (mainly below
 m
and above T g
) have been calculated following the algorithm proposed by Giordano and Dingwell (2003a). Iron has been considered to be fully trivalent and water amount always set to 0. The experimental and theoretical viscosity-temperature data have been then interpolated by a polynomial function on the form of the classical VFT equation (Giordano and Dingwell, 2003a and references therein). The viscosity at the glass transition has been fixed at 10
12
Pa s, even if this parameter is also moderately and kinetically dependent from the applied cooling rate (Debenedetti and Stillinger, 2001; Del Gaudio et al., 2007 and references therein). The temperature at which the viscosity at the glass transition is attained is T g
(Table 1). The reduced glass transition T rg
parameter has been obtained for each composition by T g
/
 m
. The variation of SiO
2
with the calculated T of melting, glass transition, and reduced glass transition is summarized in Fig. 2. As expected, the relationship between the silica content and T g
is more scattered, being the viscosity significantly correlated to the bulk composition of a silicate melts (Hui and Zhang, 2007). On the other hand, the melting temperature of a silicate melt is mainly related to the SiO
2
content. The reduced glass transition is linearly and pretty related with the silica content (Fig. 2).
Determination of NBO/T and F(kinetic fragility)
An useful parameter to correlate the chemical and physical parameters of silicate melts is
NBO/T. The polymerization of a melt may be quantified using the NBO/T calculation (e.g. Mysen
1987 and 1988). A decrease of the NBO/T means an increase of the bulk polymerization of the melt. In the following, we briefly summarise the calculation procedure and its meaning. The silicate melts are liquid ionic solutions composed of anionic clusters (or polymers) sharing exchangeable cations. The anionic clusters are dominated by tetrahedrally coordinated cations (T) because of their high field strength (charge/radius, Z/r). In the network of SiO
4
4-
tetrahedra, Si
4+
, which is the dominate cation, may be partially replaced by other cations that have a slight larger ionic radius, such as Al
3+
, Fe
3+
, B
3+
, and P
5+
. However, this substitution requires a charge-balance by association with other cations to compensate the electrical charge of about 4 (e.g. alkali or alkaline earth cations). The configurations of the anionic cluster in silicate melts are of 4 different types TO
2
, TO
3
, T
2
O
5
and TO
4
according to the number of non-bridging oxygens (NBO), oxygens unbonded to tetrahedrally coordinated cations and associated with the presence of network modifier cations (M), which increases from 0 to 4. The ratio of non-bridging oxygens to tetrahedral cations (NBO/T) is a measure of the population distribution of anionic clusters existing in a silicate melt and its variation is a function of the bulk composition. The
NBO/T ratio may be represented as the reaction between a metal oxide and a tetrahedrally coordinated cations: a) M-O-M + T-O-T = 2 · M-O-T, b) O
2-
+ O
0 = 2 · O - and c) free oxygen + bridging oxygen = non-bridging oxygens. By converting the wt% of chemical analyses of the melt in atomic proportions, it is possible to calculate the NBO/T ratio. The tetrahedral coordination is considered as the sum of Si,
Al, and Fe
3+
and alkalis and alkaline earths are assigned to A1
3+
and Fe
3+
for the charge balance. The final equations are derived by adopting the charges of tetrahedral cation (4+) and oxygen (2-): NBO =
(2O-4T) and NBO/T = (2O-4T)/T.
The fragility term of a liquid producing a glass is a measure of the deviation from an Arrhenian behaviour (Angell, 1995; Debenedetti and Stillinger, 2001; Giordano and Dingwell, 2003b). An alternative view of the fragility term is provided by a T g
-scaled Arrhenian plot; strong liquids follow a

6 linear trend, whereas fragile liquids follow a curved upward slowing down (Angell, 1995; Debenedetti and Stillinger, 2001). Silicate melts are relatively strong. Among the silicate compositions, silica is the strongest compound, whereas addition of alkalis or other divalent cations tend to increase the fragility of a silicate melt. Fragility (thermodynamic or kinetic) can be measured in several ways. A useful method is provided by Giordano and Dingwell (2003b) for magmatic silicate liquids. On the base of a large viscosity dataset, Giordano and Dingwell (2003a) were able to retrieve a relationship between
NBO/T parameter and the kinetic fragility term F . In Table 1, the F parameter for all the selected compositions has been calculated according to F = -0.0044 + 0.6887 (1 – e
-5.4767 NBO/T
) (Giordano and
Dingwell, 2003b). Figure 3 summarizes the calculated non-linear relationships between NBO/T and
SiO
2
, T rg
and F .
Estimate of the critical cooling rate
We determine the critical cooling rate R c
of the selected magma compositions following the analytical approach developed by Fan et al. (2007). They reported the measured values of T rg
and R c
in network, metallic, and molecular glasses and proposed a new dimensionless parameter
Ф
, accounting for both T rg
and the fragility index D. This
Ф
parameter is defined as T rg
(Δ T x
/ T g
) a
. To calculate
Ф,
one should also known the parameter Δ T x
= T x
- T g
; T x is defined as the (subliquidus ) temperature where the nucleation initiates by heating a glass from the glass transition region with a typical heating rate of
0.333 K/s. The measured exponential a parameter is between 0.181 and 0.143 (Fan et al., 2007). The experimental Δ T x is also a measure of the nucleation behaviour; it is linked to the fragility (D) of a liquid, being nucleation mainly affected by the viscosity at intermediate undercooling. To summarise,
GFA of a liquid at supercooling condition is sensitive to the viscosity close to
 m and T g
, but also to the viscosity at an intermediate thermal range, where the nucleation rate of any silicate melt has its maximum ( I max
and
 min
at
 max
); the viscosity around
 max
is obviously related to the fragility of a melt
(Fan et al., 2007). Silicate melts have significant fragility variations, as shown Table 1 (Angell, 1995;
Debenedetti and Stlilinger, 2001; Giordano and Dingwell, 2003b). Probably, the overestimation or underestimation of the R c
for all glasses reported by Fan et al. (2007) is due to differences in the fragility behaviours. For natural silicate melts with a similar T rg
, the supercooled liquid with the higher fragility (Tab. 1 and Fig. 3) is expected to nucleate easier with respect the stronger melt.
We fit only the experimental data relating T rg
and R c
for the network glasses reported in Fan et al. (2007) (Fig. 4). The results from the experiments are empirically related by:
R c
= 1.75·10
-7 ·
T rg
-29.6
(1)
This equation well fits the whole data set for network glasses within ±1 order of magnitude.
Using equation 1, we determine R c of the selected magma compositions reported in Tab 1 by substituting the appropriate T rg
values. The results are listed in Table 2.
DISCUSSION
The equation 1 for network silicate melts and glasses can be validated independently using the nucleation theory. Fokin et al. (2003) rearranged several equations from the classical nucleation theory
(CNT) for silicate melts with T rg
values between 0.5 and 0.6 to show as the induction time
 min
and I max

7
(both at
 max
) are related to T rg
in a homogeneous/internal precipitation process. They demonstrated remarkable correlations using experimental data and theoretical models. In Fig. 5, the experimental data used by Fokin et al. (2003) are reported. As T rg
increases, the maximum homogeneous nucleation rate ( I max
) decreases and the related induction time (
 min
) increases. Therefore, complementary approaches to estimate GFA of a silicate melt by independent parameters (i.e., R c
(Fig. 4) I max
and
 min
(Fig. 5)) are self-consistent and related to a single parameter, i.e.
T rg
. As a corollary, silicate melts cooled from (or above) the melting temperature or heated from (or below) the glass transition with low
T rg
nucleate (exponentially) more crystals in a shorter time with respect to another silicate melt with a higher T rg
.
The master trend (equation 1) reported here must be also verified and possibly constrained by available solidification data produced in laboratory or inferred by natural cooling data for fully glassy lavas. It is important to stress that the relation (1) is obtained for network glasses (mainly silicates) in which the cooled liquid solidifies a single crystal phase with the same chemical composition of the melt. This feature contrasts with the crystallization of natural silicate melts, which involve the nucleation of several phases inducing continuous chemical changes in the liquid fraction as a function of the progressive crystallization or dissolution.
As already stated, relatively few investigations are reported in the petrological, geochemical and volcanological literature on GFA and the nucleation behaviour. In the following, we analyse all the bibliographic data dealing on the solidification of several anhydrous to quasi -anhydrous silicate melts and compare these data with those from the our theoretical model.
Theoretical vs experimental R c
Rhyolitic melts . A summary on the cooling behaviour of rhyolitic melts is reported in Swanson et al. (1989). Experimental cooling and heating data on similar silicate compositions were compared with natural petrographic features (Lofgren, 1971; Swanson, 1977; Naney and Swanson, 1980). In this study, the textural features of two main drill holes, one located in the near vent zone and one in the distal zone (labelled 2a and 2b) of the Obsidian Dome (Inyo domes, California), were analysed as a function of the depth of samples in the two holes. From a volcanological point of view, the two drill holes include (from top to bottom) a pumice layers (from 12 to 18 m) and a glassy-aphyric to a holocrystalline devitrified lava(s) (from 30 to 37 m). The obsidian occurs only in the frontal zone of the dome. The low amount of large phenocrysts (lesser than 10 vol.%, with an average of less than 2 vol.
%) was interpreted as an evidence of magma erupted close to the liquidus temperature. The difference in microlite amount was mainly attributed to differences in both volatile content (<0.5 wt.%) and bubbles, excluding the effects of devitrification occurred in the interior part of the dome (drill hole 2a).
The exterior part of the Obsidian Dome was instead considered to be anhydrous (lesser than 0.1 wt.% of volatile; Swanson et al., 1989). The obsidians cropping out in the distal areas of the dome have been analysed to obtain crystal fabric information by Castro et al. (2002); 13 microlite poor obsidian samples at the base of the flow were considered. The amount of these microlites is lesser than 4 vol.%. Castro et al. (2002) used the recovered strain data to infer an emplacement time of about 400 days for the
Obsidian Dome. Hence, it is possible to calculate a cooling of 400 °C in about 3.4 10
7
s (400 days), being the initial temperature near T m
(about 1160 °C) and the glass transition T g
at ~760 °C. The inferred critical cooling rate is close to 2 10
-5
K/s and the calculated R c
is 6 10
-3
K/s (Table 3). If part of the microlites crystallised during the obsidian lava flow emplacement, it is possible to infer a critical

8 cooling rate higher than 2 10
-5
. A higher cooling rate can be also attributed to a shorter emplacement time.
Other examples of rhyolitic lavas for which (partial) cooling data are available are the Rocche
Rosse obsidian flow, Lipari Island, Italy (Gottsman and Dingwell, 2001a) and the peralkaline rhyolite
(pantellerite) lava from Mayor Island, New Zealand (Gottsman and Dingwell, 2002b). In the first study, several fresh aphyric samples from the frontal ramp of were sampled and analysed by differential scanning calorimetry (DSC) to derive the cooling history of the lava across the glass transition region.
Being the lava phenocryst-free, it can be retained that the cooling regime started near or above T m
and the cooling rate was probably not so different from that measured in the glass transition region.
Gottsmann and Dingwell (2001a) retrieved cooling rates ranging between 0.003 and 0.00064 K/s for these obsidian samples as a function of their position in the flow front. The value of R c
calculated with the equation 1 for this flow is 0.00026 K/s, a value close to the lower cooling rate measured by relaxation geospeedometry (Table 3). The obsidian samples of Mayor Island have a lower estimated cooling rate equals to 0.000012 K/s. Again, the cooling rates were estimated by DSC measurements in the glass transition region (Gottsman and Dingwell, 2002b). The crystal amount in the investigated glassy samples by DSC is lesser than 1 vol. %. The calculated R c for this peralkaline melt is 0.013 K/s
(Table 3), that is significantly higher than the measured lower cooling rate. However, Gottsmann and
Dingwell (2002b) also reported a maximum cooling rate of 0.1 K/s. The geospeedometry method using the relaxation of a glass reheated in the glass transition interval is able to account of cooling data only in the thermal region where a lava is approaching a brittle behaviour. In any case, for a phenocryst-free lava it can be reasonably assumed that the heat dissipation is roughly linear from high to low temperatures, especially for obsidians where crystallization does not occur or is limited, as in the two obsidians described above.
Basaltic melts . The solidification and the crystallization behaviour of basaltic melts have been investigated by Lofgren (1980) and Lasaga (1997 and reference therein). However, only few investigations are available for basaltic melts cooled from or above the liquidus temperatures down to the solidus or glass transition. Moreover, the applied cooling rates used in the experimental studies are relatively low, in the order of less than 1 K/s. Indeed, supercooled basaltic lavas have been observed directly to cool with a rate as high as 240 K/min or 6 K/s, or, more commonly, with a rate of about 170
K/min or 2.8 K/s (Gottsmann et al., 2004). These values are also in the order of the revisited determination of critical cooling rates derived by Yinnon et al. (1980) for several lunar basalts with composition (wt.%): 44.2<SiO
2
<48.2, 7.4<Al
2
O
3
<26.7, 4.9<FeO<20.3, 5.4<MgO<18.1, 8.2
<CaO<16.1, 0.2<alkalis<0.9. For these melts, the critical cooling rates based on theoretical and experimental data were inferred to range between 0.11 and 8.3 K/s. The highest cooling rate was found for the Fe- and Mg-rich compositions. Using equation (1), a R c
of about 15 K/s implies a reduced glass transition of 0.54, whereas a R c
of about 142 K/s implies a T rg
of 0.5. One basalt in Table 1 (sample
STB) agrees well with this cooling rate, whereas the other two basalts require critical cooling rates in the order of 10
3
(sample SLP) to 10
4
K/s (sample peridotite). We emphasise that this latter basalt is extremely Mg-rich (Table 1). Several Mg-rich basic compositions similar the peridotitic sample of
Dingwell et al. (2004) were also investigated from temperature slightly above the liquidus to 1200 °C by Lofgren (1989). The highest cooling rate used in that study was 0.51 K/s, producing about 80 vol.% of crystals. Hence, the calculated cooling rate (~ 10
4
K/s) is very high, but the high crystal content from the dynamic cooling experiment(s) of Lofgren (1989) clearly indicates that a higher cooling rate
(>>0.51 K/s) is required to suppress the nucleation process. Conte et al. (2006) also investigated the

9 crystallization dynamic of a relatively evolved basalt, which is cooled from near liquidus temperature
(1200 °C) to 1100 °C with two different cooling rates (Table 3). For the higher imposed cooling rate of
0.25 K/s, at 1150 °C the run-product resulted to contain 23 vol.% of crystals. As in the previous example, this suggests a cooling rate significantly higher than that measured to prevent crystallization.
We estimate a value R c
from T rg
of about 13 K/s (Table 3). A basaltic andesite silicate melt slightly more evolved than the previous one was also investigated by Conte et al. (2006) under the same cooling condition of 0.25 K/s between 1200 and 1150 °C. This basaltic andesite crystallised lesser than
10 vol.%. This suggests that the GFA of this composition is higher than the sample PST9, according to the calculated R c
of 5.53 K/s, which, in turn, reflects a higher melt polymerisation, as reported in Table
3.
Intermediate silicate melts . Recent studies dealing on the solidification of slightly alkaline trachytic and latitic melts (Iezzi et al., 2008) and on two andesitic liquids by Mollo et al. (2007)
(preliminary results) are described in this section. Their features are reported in Table 3. All these four silicate melts were cooled from a superheating temperatures of 1400 °C for 20 minutes and, successively, the temperature was lowered at 1300°C in five minutes. Such a superheating treatment should reduce the possible heterogeneous effect on successive nucleation behaviour in the supercooled region (see below). From 1300 to 800 °C, five different cooling rates were applied, equal to: 0.417,
0.208, 0.050, 0.008 and 0.002 K/s. The results of these experiments were that the two andesites started to crystallise with a 0.417 K/s cooling rate, whereas latite nucleated only below 0.050 K/s and trachyte never produced any crystal phases under these cooling regimes. The calculated and observed R c
values of one andesite are in good agreement (Table 3), whereas those of latite and the other andesite differ of
27 and 20 times, respectively. For the trachyte, the difference is on in order of 500 times (Table 3).
Remarkably, the T rg
of these melts are roughly similar among them (Table 3), except that andesites have a significant higher fragility behaviour compared to latites and trachytes. Probably, albeit the differences in F are not so high (Table 3), the stronger behaviour of the two latter melts contributed to increase the kinetics of nucleation.
Alkaline to peralkaline silicate melts . Solidification, nucleation and crystallization features of these silicate liquid compositions have been not investigated until now. The only data in our knowledge are reported by Gottsmann and Dingwell (2001b) for a phonolite obsidian form Tenerife (Canary
Islands). The composition of one of these samples is reported in Table 1 (sample MB5). The basal poorly vesicular obsidian flow has been inferred to experienced a cooling regime in the glass transition region as low as 0.00007 K/s; other samples from the MB5 deposit gave cooling rates higher than
0.006 K/s. These former obsidian phonolites have sparse phenocrystals, whereas the amount of microlite is between 5 and 20 vol.%. The microlites amount increases as the volatile content of the melt(s) also increases (Gottsmann and Dingwell, 2001b). The required cooling rate, which we assume to be constant for the entire cooling history of the flow, is, on average, of 10
-4
K/s. The calculated R c using the determined T rg
of 0.57 (Table 1) gives 2.9 K/s (Table 2). The marked discrepancy of R c values is, however, probably lower if we consider the amount of microlites in this phonolite.
Assessment of the Glass Forming Ability
An relevant aspect in the assessment of GFA of a silicate melt is the superheating effect on nucleation (Lofgren, 1980; Kirkpatrick, 1981). Only few studies have experimentally investigated this effect on the cooling rate starting from temperatures significantly above the melting temperatures down

10 to solidus or glass transition (Mollo et al., 2007; Iezzi et al., 2008). This effect was previously ascribed to the local molecular readjustment induced by thermal variations (cooling or heating) of a melt above its liquidus as a function of time (Kirckpatrick, 1981; Swanson et al., 1989). However, it is now well established that the short-range order (atomic and molecular) rearrangements or the relaxation of any silicate melt is extremely rapid (lower than seconds) above the glass transition region. This relaxation time is defined by the Maxwell relation:
τ
=
η
/ G , where
η
is the viscosity of the melt and G is the shear modulus; the variations of
η
and G at high temperature impose that the relaxation time is in the order of microseconds (Dingwell and Webb, 1990; Webb, 1997). Recently Pupier et al. (2007) suggested that the superheating effect, which they observe in dynamic cooling experiments, can be attributed to the presence of heterogeneous nucleation sites during the superheating treatment of the melt. They proposed the presence of gas bubbles as preferential sites of nucleation. Alternatively, it can be also suggested that small and few crystals can be metastably present above the melting temperature of a silicate melt. These preferential sites of nucleation require a certain time to be fully melted, depending on the temperature and time spent in the superheating region. This suggestion appears agrees well with the data of Tsuchyama (1983). This author reported that the superheating effect on nucleation is lacking for a temperature of 40 °C above the liquidus (regardless the time duration) or, alternatively, for any virtual superheated temperature for longer experimental time. In both these two hypotheses, the experimental data attempting to directly investigate these phenomena are, however, not available. Such investigation would be highly desirable in the future. Here, it is only possible to remark that silicate melts of rhyolitic, dacitic, trachytic, latitic and andesitic composition (Swanson, 1977;
Fenn, 1977; Naney and Swanson, 1980; Mollo et al. 2007; Iezzi et al., 2008) are able to nucleate internally at random and isolated zones and preferential nucleation on sample containers (mainly Pt capsules or wires) are not detected. These features contrast with the common observations on basaltic melts solidified in laboratory, for which a marked effect of the sample holder on the nucleation behaviour is frequently reported (Dowty, 1980; Lofgren, 1980; Berkebille and Dowty, 1982; Pupier et al., 2007). Also, it is noteworthy that evolved silicate melts as natural rhyolites (Swanson et al., 1989), or experimentally crystallised rhyolites, dacites and latites (Naney and Swanson, 1981; Iezzi et al.,
2008) stabilised crystal phases that are far from the equilibrium phases. All these studies also reported that the early phases are Fe-Ti oxides and pyroxenes and not feldspars, as expected. A similar result was reported by Sharp et al. (1996), which show disequilibrium pyroxene microlites and “nanolites” stabilization by SEM and TEM observations, in a natural rhyolitic dome. All these kinetic features can be qualitatively explained by the theory of nucleation proposed by Kirkpatrick (1983) for silicate melts.
The activation energy required for the onset of nucleation is proportional to the number of tetrahedral units. A high number of tetrahedral units and corner connections (either in the silicate melt and growing crystal phase), measured by the NBO/T parameter, imply a high nucleation barrier, which, in turn, determines high incubation times. Therefore, a silicate melt and/or crystal with a relative low number of tetrahedral units (low NBO/T) and degree of polymerization has a lower incubation time with respect to more polymerised and T-O rich (high NBO/T) melts and crystals. Furthermore, between two stable or meta-stable crystal phases in a melt, the phase with the lower number of tetrahedral units and polymerisation degree will nucleate faster (metastable nucleation).
The effects of superheating also reduces the kinetic effect of nucleation and, consequently, reduces GFA of a silicate liquid. However, if this effect is independent from the melt composition, the reduced glass transition should be re-scaled. By contrast, if the superheating effect on the nucleation is a function of the melt composition, as it seems from the above discussion, then, the lesser evolved

11 natural anhydrous silicate melts will be more significantly affected by the presence of heterogeneous nucleation-induced sites.
Another important parameter controlling the nucleation behaviour of a liquid during cooling is the dynamic regime. Kouchi et al. (1986) verified experimentally the effect of stirring on a basaltic silicate melts. They found that the nucleation density strongly increases as the applied stress rate increases. Effects of mechanical perturbation on more evolved silicate melts are not known.
CONCLUSION
The results of our model and those inferred from available experiments show that the theoretical determination of R c
by T rg
for natural anhydrous silicate melts generally holds within two orders of magnitude. An exception to this rule has been detected for trachyte (Iezzi et al., 2008) and for the lesser evolved Mg-rich basalts. The R c
values determined using T rg
invariably tend to be higher of those obtained from experiments, e.g. the andesites of Mollo (2007) and the latite and trachyte of Iezzi et al.
(2008), in which a direct comparison between experimental and theoretical data is fully justified.
Although the limitations of the few available experimental data are significant, however, these data show that to vitriphy a completely melted basalt ( liquidus to superheated temperatures) a cooling rate of 0.1-0.5 K/s is not sufficient, whereas a rhyolitic glass can be obtained with cooling rates lower than 0.001-0.0001 K/s. Intermediate sub-alkaline melts (basaltic andesites, andesites and latites) are able to produce a glass with cooling rate values between those required for basalts and rhyolites.
According to the R c
values inferred from the equation 1, dacitic melts can be expected to vitriphy with dynamic coolings in the order of 0.01-0.001 K/s. Therefore, natural basic melts crystallised more rapidly than the silicic ones. This also the behaviour of simple, synthetic silicate melts (Fokin et al.,
2003; Fan et al., 2007). Between two sub-alkaline melts with a difference in T rg
of several unit percent, the melt with the lower T rg
has a lower GFA. According to the general relations proposed by Fokin et al. (2003) and summarized in Fig. 5, the GFA of two melts with comparable T rg
cannot be easy estimated. However, small differences in T rg
can be related to a relative low accuracy in the determination of
 m and/or T g
. For instance, a thermal inaccuracy of only ± 10 °C implies an error on the determination of the critical cooling rate of 4 times, whereas an incertitude of ± 20 °C determines a variability of 16 times. If the kinetic fragility of all the silicate melts reported in Fig. 1 and Table 1 could be taken in account, the calculated R c of the lesser evolved liquids should be even higher than that calculated with equation 1 (Fan et al., 2007). This because as shown in the fragility linearly increases as the reduced glass transition decreases (Fig. 6).
An outcome of our study is that for R c
values of 0.1-0.5 K/s, an initially glassy anhydrous basaltic lava may change the rheology from Newtonian to Bingham due to crystallization, whereas a rhyolitic flow will be mainly characterized by a Newtonian behaviour down to the glass transition region because it remains crystal-free for a longer time. In addition, in an anhydrous basaltic lava with
R c
~ 0.1-0.5 K/s, the viscosity will depend on temperature, crystal content and residual melt composition. For a same cooling rate and anhydrous conditions, the viscosity of rhyolitic or trachytic lavas will largely depend on the temperature because crystallization processes are not easy to develop.
With respect to basaltic lavas, flows of evolved composition will have more chance to remain glassy.
Fully glassy subaerial obsidian-like lava flows have consistently rhyolitic or trachytic composition. At comparable cooling rates, a more silicic lavas will flow for longer time with respect to less evolved

12 flows, which progressively increase their crystal content. This is consistent with observations of lavas and domes (Westrich et al., 1988; Swanson et al., 1989; Sparks et al., 2000).
Further studies are required to elucidate the role of water, superheating temperature, and strain rate on GFA of magmatic melts. In addition, experimental data are required to increase the accuracy in the estimate of GFA. The role of heterogeneous nucleation must also be investigated at subliquidus conditions, being terrestrial lavas normally erupted with a certain amount of phenocrysts acting as centres of preferential nucleation.
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16
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17
TABLE CAPTIONS
Table 1.
Sample labels and reference, classification, chemical compositions, petrological and rheological parameters of the selected natural melts. The reference are: 1 – Dingwell et al. (2004); 2
- Whittington et al. (2001); 3 - Gottsmann et al. (2002); 4 - Richet et al. (1996); 5 – Giordano and
Dingwell (2003a); 6 – Giordano et al. (2006)
Table 2.
Calculated values of R c, according to equation 1 for the compositions reported in Table 1.
Table 3.
Composition and rheological parameters of selected silicate melts. Rc values can be inferred from experimental results (Reference).

Table 1
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
3
4 reference
1
2
2
3
3 label peridotite
SiO
2
TiO
2
Al
2
O
3
45.83 0.18 4.87
FeO tot MnO MgO CaO Na
2
O K
2
9.58 0.00 31.63 6.37
O P
2
O
5
Tot alkali
0.33 0.00 0.00 98.80 0.33
T.A.S. basalt
NBO/T F T m
(°C) T g
(°C) T rg
1.6 0.68 1600 680 0.43
Trachite 64.44 0.50 16.71 0.00 0.00 2.92 5.36 6.70 3.37 0.00 100.00 10.07
Phonolite 58.82 0.79 19.42 0.00 0.00 1.87 2.35 9.31 7.44 0.00 100.00 16.75 trachyte phonolite
0.22 0.48 1160 690 0.59
0.2 0.46 1200 650 0.54
R839.5.8 52.12 2.61 13.69 11.06 0.16 6.37 11.00 2.33 0.41 0.25 100.00 2.74 basaltic-andesite 0.33 0.57 1164 575 0.49
MB5 60.68 0.56 18.88 3.32 0.20 0.36 0.68 9.79 5.47 0.06 100.00 15.26 phonolite 0.05 0.16 1039 595 0.57
P3RR 75.37 0.07 12.84 1.57 0.07 0.04 0.72 4.19 5.13 0.00 100.00 9.32
ME1311e 59.70 0.52 18.30 7.58 0.20 2.59 7.21 3.44 0.90 0.17 100.61 4.34
EIF
ETN
41.15 2.74 12.10 10.11 0.00 11.24 15.66 2.76 3.04 1.02 99.82 5.80 rhyolite andesite basanite
47.03 1.61 16.28 10.13 0.20 5.17 10.47 3.75 1.94 0.59 97.17 5.69 trachy-basalt
0 0 989 655 0.66
0.04 0.14 1193 740 0.62
0.88 0.68 1355 620 0.46
0.29 0.54 1166 650 0.56
Ves_G_tot 49.20 0.83 16.40 7.20 0.13 5.10 10.20 2.70 6.50 0.72 98.98 9.20 phono-tephrite 0.34 0.58 1258 670 0.53
N_An 61.17 0.84 17.29 5.39 0.00 3.35 5.83 3.85 1.39 0.00 99.11 5.24 andesite 0.09 0.27 1168 715 0.61
Ves_W_tot 51.20 0.67 18.60 6.10 0.13 2.50 7.30 3.75 7.90 0.40 98.55 11.65 tephri-phonolite 0.18 0.43 1269 650 0.51
W_Tph 50.56 2.35 14.03 0.00 0.00 8.79 15.00 7.04 3.01 0.00 100.78 10.05 tephri-phonolite 0.94 0.68 1198 660 0.55
UNZ 66.00 0.36 15.23 4.08 0.10 2.21 5.01 3.84 2.16 0.14 99.13 6.00 dacite 0.08 0.23 1131 685 0.61
Ves_W 52.02 0.59 19.28 4.65 0.14 1.72 6.58 4.53 7.69 0.65 97.85 12.22 tephri-phonolite 0.15 0.37 1269 640 0.5
Ves_G 51.24 0.58 19.14 4.55 0.12 1.71 6.51 4.60 7.99 0.71 97.15 12.59 tephri-phonolite 0.15 0.39 1258 650 0.52
ATN 59.70 0.46 18.52 3.60 0.17 0.65 2.80 3.89 8.45 0.15 98.39 12.34 trachyte 0.02 0.08 1178 690 0.59
AMS_B1 60.10 0.38 18.03 3.43 0.14 0.73 2.92 4.49 7.89 0.16 98.27 12.38
AMS_D1 59.98 0.39 18.01 3.82 0.11 0.88 2.91 4.06 8.37 0.21 98.74 12.43
MNV 63.88 0.31 17.10 2.90 0.13 0.24 1.82 5.67 6.82 0.05 98.92 12.49
PVC 63.99 0.45 16.96 2.55 0.14 0.32 0.83 6.33 6.37 0.09 98.03 12.70
HPG8 78.60 0.00 12.50 0.00 0.00 0.00 0.00 4.60 4.20 0.00 99.90 8.80 trachyte trachyte trachyte trachyte rhyolite
0.04
0.04
0.02
0.02
0
0.14
0.13
0.08
0.06
0
1152
1150
1069
1045
1015
695
685
665
690
830
0.59
0.56
0.59
0.64
0.8
STB
MRP
49.07 0.98 16.91 8.36 0.22 5.73 10.88 2.63 2.20 0.00 96.98 4.83 basalt 0.3 0.55 1188 620 0.52
53.53 0.82 18.95 9.03 0.19 3.42 9.23 3.45 1.64 0.00 100.26 5.09 basaltic-andesite 0.12 0.33 1213 695 0.57
SLP 45.76 2.27 12.52 11.30 0.25 11.42 11.45 2.65 1.07 0.86 99.55 3.72
CLOF 68.80 0.23 12.58 3.17 0.14 1.24 3.43 4.01 6.18 0.03 99.81 10.19 basalt trachyte
0.62
0.11
0.66
0.31
1511
1038
660
680
0.44
0.66
Fra 55.41 0.72 18.38 7.31 0.16 2.39 5.76 4.23 4.58 0.00 98.94 8.81 trachy-andesite 0.08 0.24 1142 700 0.61
MST 60.71 0.58 18.29 6.38 0.19 2.58 7.10 3.57 0.85 0.00 100.25 4.42 andesite 0.06 0.18 1198 695 0.58
18

Table 2
Reference
1
5
6
6
5
5
5
5
5
5
5
5
5
6
6
6
6
5
5
5
5
5
3
3
4
2
2
3
T.A.S. SiO
2
Log ( basalt 45.83 trachyte phonolite
64.44
58.82 basaltic-andesite 52.12 phonolite rhyolite andesite
60.68
75.37
59.70 basanite 41.15 trachy-basalt 47.03 phono-tephrite 49.20 andesite 61.17 tephri-phonolite 51.20 tephri-phonolite 50.56 dacite 66.00 tephri-phonolite 52.02 tephri-phonolite 51.24 trachyte 59.70 trachyte 60.10 trachyte trachyte trachyte
59.98
63.88
63.99 rhyolite basalt
78.60
49.07 basaltic-andesite 53.53 basalt trachyte
45.76
68.80 trachy-andesite 55.41 andesite 60.71
R c
) (K/s)
0.69
0.06
-1
-4.37
1.53
0.48
0.9
-0.36
1.95
1.53
0.06
0.06
2.8
0.69
1.32
-0.36
1.74
3.43
0.06
1.11
2.17
0.48
-1.42
-0.58
3.22
-1.42
-0.36
0.27
19

Table 3
Reference composition
CaO
BaO
Na
2
O
K
2
O
P
2
O
5
F
SiO
2
TiO
2
Al
2
O
3
Fe
2
O
3
MnO
MgO
Cl
SO
3 total
NBO/T
F
T m
(°C)
T g
(°C)
T rg
Estimated Rc (K/s)
Calculated Rc (K/s)
Swanson et al., 1989
Rhyolite
(horneblendebearing)
4.56
4.85
0.09
-
-
-
98.95
0.01
69.42
0.34
14.99
2.76
0.06
0.42
1.46
-
0.00
1160
760
0.655
> 0.00002
0.006
Swanson et al., 1989
Rhyolite
(horneblendefree)
72.95
0.14
14.20
1.76
0.06
0.01
0.80
-
4.35
5.28
0.02
-
-
-
99.57
0.01
0.00
1160
765
0.659
> 0.00002
0.006
Gottsmann and
Dingwell, 2001
Rhyolite
(sample P3RR-33)
4.13
5.03
0.01
0.14
0.29
-
100.61
0.01
75.55
0.06
12.83
1.78
0.06
0.04
0.69
-
0.00
985
770
0.782
0.003-0.0005
0.00026
Gottsmann and
Dingwell, 2002
Peralkaline rhyolite
6.15
4.20
0.01
0.14
0.20
-
100.95
0.02
74.11
0.23
9.54
6.03
0.13
0
0.21
-
0.01
935
640
0.684
0.1
0.000012
0.013
Conte et al.,
2006
Basalt
(sample PST9)
2.32
1.88
0.47
-
-
-
100.25
0.44
50.49
0.86
15.38
8.47
0.21
7.69
12.48
-
0.63
1200(*)
650
0.542
>0.25
13.32
Conte et al.,
2006
Basaltic andesite
(sample ST17)
2.27
1.62
0.22
-
-
-
100.16
0.29
55.49
0.68
15.04
8.23
0.16
6.43
9.68
-
0.55
1200(*)
670
0.558
>0.25
5.53
3.9
1.9
-
-
-
-
100.4
0.23
0.50
1240
690
0.556
< 0.4167
> 0.208
6.003
57.1
-
19.8
3.2
-
4.5
10.0
-
2.33
1.85
0.16
-
-
-
99.79
0.11
0.31
1220
720
0.590
< 0.4167
> 0.208
1.053
58.73
0.64
17.10
7.79
0.14
3.77
7.48
-
Mollo et al.,
2007
Andesite
(sample
DPP)
Mollo et al.,
2007
Andesite
(sample
GI1)
Iezzi et al., 2008
Trachyte
3.72
7.90
0.19
0.16
0.21
0.03
99.61
0.07
57.72
0.39
18.40
5.56
0.10
1.46
4.23
0.09
0.21
1150
680
0.591
< 0.0021
0.994
Iezzi et al., 2008
Latite
4.39
5.18
0.35
0.15
0.03
0.01
99.72
0.09
55.36
0.60
18.40
7.07
0.10
2.21
5.54
0.14
0.27
1180
690
0.585
< 0.05 and >
0.008
1.383
20

21
FIGURE CAPTIONS
Figure 1.
SiO
2
vs. Na
2
O+K
2
O diagram (TAS) of the selected melt compositions.
Figure 2.
Plot of SiO
2
content vs. (a) calculated T of melting, (b) T of glass transition at 10 12 Pa s, and (c) reduced glass transition parameter. The solid lines are linear fits of the data; the dotted lines represent the 95% confidence interval.
Figure 3.
NBO/T vs. (a) silica content, (b) Trg , and (c) kinetic fragility F . The solid lines are polynomial fits of the data and exponential fit using the relation by Giordano and Dingwell (2003b).
Figure 4.
Relationships between T rg
and R c
for the silicate compositions (network glasses) from the experimental data of Fan et al. (2007).
Figure 5. Experimentally determined relationships between T rg
, I max
(circles) and
 min
(crosses) from
Fokin et al. (2003).
Figure 6.
Relation between the calculated T rg
and the kinetic fragility F . The line is the linear best fit (R 2 =0.6).